The IceCube apparatus can measure directions of neutrinos/antineutrinos
[1]}. The IceCube experimental data discussed below
is selected
taking into account only such \(\nu _{\mu }\)  neutrinos that passed through
the Earth (see Fig. REF ).
In Fig. REF  we present our results for the atmospheric
\(\nu _{\mu }\)  flux, scaled by a factor \(E_{\nu }^2\) , which is the sum of
the conventional and prompt contributions. The predictions were obtained
considering different models for the uPDFs and distinct values for
\(P_{ic}\)  in the calculation of the prompt contribution. Moreover, for
the conventional atmospheric neutrino flux we assume the result derived
in Ref. [2]}. The resulting predictions are compared with
the IceCube data obtained in Ref. [3]} for the zenith-averaged flux of atmospheric neutrinos. For completeness, the
results from Ref. [4]} for the astrophysical neutrino
flux are represented by the grey band.
One has that the prompt contribution enhances the flux at large
neutrino energies, with the enhancement being strongly dependent on the
magnitude of the \(cg \rightarrow cg\)  mechanism and the uPDF considered as
input in the calculations. If this mechanism is disregarded, the results
represented by “Conv. + gg” in the figures indicate that the impact
of the prompt flux is small in the current kinematical range probed by
IceCube. In particular, it is negligible when the saturation effects are
taken into account [see Fig. REF  (b)].
On the other hand, the inclusion of the \(cg \rightarrow cg\)  mechanism
implies a large enhancement of the prompt flux at large \(E_{\nu }\) , with
the associated magnitude being strongly dependent on the value of
\(P_{ic}\) . Our results for the KMR uPDF, presented in Fig. REF  (a), indicate that a value of \(P_{ic}\)  larger than \(1.5 \%\)  implies a prediction
for neutrino flux that overestimate the IceCube data at high energies.
We have verified that a similar result is obtained for the KS linear
uPDF (not shown explicitly). Therefore, the results derived assuming
that the QCD dynamics is described by linear evolution equations,
which disregard the saturation effects, indicate that in order
to describe the current IceCube data we
should have \(P_{ic} \lesssim 1.5 \%\) . Surely, future data can be more
restrictive in the acceptable range of values for \(P_{ic}\) . In
contrast, the results presented in Fig. REF  (b) suggest
the presence of saturation effects with \(P_{ic} = 1.5\%\)  is not
discarded by the current IceCube data.
It is important to emphasize that the values of the \(P_{ic}\)  probabilities suggested above could be slightly
decreased if a smaller value of the \(p_{t0}\)  parameter was used
in the numerical calculations of the \(cg^* \rightarrow cg\)  cross section.
